Why the Distinction Matters
In a longitudinal study, a participant may miss a scheduled visit and return later, or may stop participating in follow-up altogether. These events can look similar in a database if both are recorded simply as “missing.” Their observed histories differ, however: a temporary absence leaves a later outcome observed, whereas permanent withdrawal produces a terminal stretch of unobserved scheduled outcomes. A correct description of the response history is essential before labeling a participant as having dropped out.
The distinction is about the pattern and the study’s follow-up status, not about why the participant was absent. A missed visit may have many causes, and an observed pattern by itself does not establish a reason. This tutorial focuses on defining and recording the distinction, not on classifying the missingness mechanism.
Visit-Level Notation and Definitions
Suppose outcomes are scheduled at visits \(j=1,\ldots,J\) for participant \(i\). Let \(Y_{ij}\) denote the outcome intended to be measured at visit \(j\), and let \(R_{ij}\) record whether that outcome is observed. The response indicator is defined as \(R_{ij}=1\) when the outcome is observed and \(R_{ij}=0\) when it is not. The response history is the ordered vector \((R_{i1},\ldots,R_{iJ})\). For example, \((1,0,1,1)\) says that visit 2 was missed, but outcomes were observed at visits 3 and 4.
The phrase “permanent withdrawal” is operational: it means that the participant will no longer contribute scheduled follow-up outcomes under the study’s follow-up plan. It does not mean the participant can never be contacted again, nor does it identify why follow-up ended. A temporary absence may include several consecutive missed visits, provided a later scheduled outcome is observed.
When response is complete through the final planned visit, there is no dropout within the planned follow-up. When follow-up ends because the study reaches its planned end, the final unobserved visits, if any, should not automatically be attributed to participant withdrawal; planned administrative censoring is a distinct reason for the observation boundary.
How the Histories Differ
A response history with a zero followed by a one cannot represent monotone dropout over the full scheduled period: the later one documents a return to observed follow-up. By contrast, a terminal sequence of zeros is consistent with dropout after the last observed visit, provided withdrawal rather than another observation boundary is documented. The sequence alone describes availability; study records are needed to establish the status that produced it.
| Response history | Observed pattern | What can be concluded |
|---|---|---|
| \((1,1,0,1,1)\) | A missed visit followed by observed outcomes | Intermittent missingness occurred; the visit-3 absence is not terminal dropout. |
| \((1,1,1,0,0)\) | Observed outcomes end before a terminal stretch of missing visits | Consistent with dropout after visit 3, if withdrawal is confirmed. |
| \((1,1,1,1,1)\) | All planned outcomes observed | No dropout within the scheduled visits. |
Worked Example: One Missed Visit, Then Return
Suppose a hypothetical study schedules five quarterly visits. Participant A has outcomes observed at visits 1, 2, 4, and 5, but not at visit 3. Write the response history and decide whether visit 3 is a dropout time.
The indicator at each visit is 1 for an observed outcome and 0 for an unobserved outcome. Thus, \(R_A=(1,1,0,1,1)\): visits 1 and 2 are observed, visit 3 is missed, and visits 4 and 5 are observed.
Because \(R_{A3}=0\) and \(R_{A4}=1\), the absence at visit 3 is followed by a later observed outcome. It is a temporary missed visit, producing intermittent missingness. Visit 3 is not a dropout time under the terminal-dropout definition: follow-up demonstrably continued to visit 4. The pattern does not, by itself, tell us why visit 3 was missed.
A useful way to check a proposed dropout label is to scan the response vector from left to right. If a zero is followed anywhere by a one, the missingness is not monotone across the scheduled visits. If the final observed outcome is at visit \(L_i\), then all later indicators are zero, so \(R_{ij}=0\) for every \(j>L_i\). That terminal shape is necessary for monotone dropout, but it is not sufficient to prove withdrawal: administrative end, death, or another observation boundary may also leave later outcomes unobserved.
Defining the Dropout Visit Carefully
Two visit indices are often useful and should not be confused. The last observed visit, \(L_i\), is the largest scheduled visit with \(R_{ij}=1\), when at least one outcome is observed. If permanent withdrawal occurs after that visit and before the next scheduled outcome, the first missing visit due to withdrawal is \(D_i=L_i+1\). Thus, the last observed visit is not itself the first dropout visit. If all visits are observed, \(L_i=J\) and there is no dropout visit within the planned schedule.
This expression for \(D_i\) applies only when follow-up ends after \(L_i\) and before the next scheduled visit, with the terminal absence attributed to withdrawal. For a participant with an earlier temporary absence, \(L_i\) alone does not identify a dropout time: the participant can have \(R_{ij}=0\) and still have a later \(R_{ik}=1\). A definition that uses only the first missing indicator would incorrectly classify that participant.
Worked Example: Locating the First Missing Visit After Withdrawal
In a hypothetical six-visit study, participant B has outcomes observed at visits 1 through 4. The participant then formally withdraws before visit 5, and no later outcomes are collected. Determine the response history, last observed visit, and dropout visit.
The response indicators are \(R_B=(1,1,1,1,0,0)\). The last observed visit is \(L_B=\max\{1,2,3,4\}=4\). Since withdrawal occurs before the next scheduled outcome, \(D_B=L_B+1=4+1=5\).
Visit 5 is the first scheduled visit with an outcome unobserved because follow-up has ended; visit 4 is the last observed visit. The visit-6 zero is part of the terminal stretch. The conclusion relies on the stated withdrawal record, not solely on seeing two zeros at the end of the vector.
Temporary Absence Can Last More Than One Visit
The word “temporary” does not require exactly one missed visit. For example, a participant might miss visits 2 and 3, then return at visit 4. The response history \((1,0,0,1,1)\) contains two consecutive missing outcomes, but visit 4 shows that the absence was not terminal. The relevant feature is whether a later scheduled outcome is observed, rather than the number of consecutive missed visits.
Worked Example: Two Missed Visits Before Resumption
Suppose a hypothetical participant has outcomes observed at visits 1, 4, and 5 of a five-visit study, but not at visits 2 and 3. The study log records that the participant returned for visit 4. Identify the missingness pattern and explain why the first zero is not dropout.
Coding each observed outcome as 1 and each unobserved outcome as 0 gives \(R_C=(1,0,0,1,1)\). The zeros at visits 2 and 3 are followed by ones at visits 4 and 5. Therefore both absences belong to an intermittent pattern, and neither is a terminal dropout time.
The return at visit 4 is decisive for the observed-history classification: \(R_{C2}=0\) but \(R_{C4}=1\), and \(R_{C3}=0\) but \(R_{C4}=1\). The pattern does not establish why visits 2 and 3 were missed; it establishes only that outcome collection resumed.
Practical Checks and Failure Modes
Before labeling a participant as withdrawn, compare the outcome table with attendance and follow-up records. In particular, verify that an apparent terminal sequence is not caused by a data-entry omission, an unscheduled visit recorded under the wrong date, or the study reaching its planned end. A later observed scheduled outcome means the earlier absence was not terminal, even if a record initially labeled the participant as lost to follow-up.
Useful checks include listing response vectors that contain a zero followed by a one, tabulating the last observed visit, and comparing terminal zero sequences with documented withdrawal dates and planned study end dates. Disagreement between these sources is a data-quality or status-definition problem to resolve, not evidence that the response history should be relabeled to fit an assumption.
For each participant and planned visit, record whether the outcome was observed: 1 or 0.
If any zero is followed by a later one, classify the pattern as intermittent for those missing visits, not as monotone dropout.
If all outcomes after a last observed visit are missing, verify whether follow-up ended through withdrawal rather than administrative study end or another boundary.
State the last observed visit \(L_i\) and, when withdrawal is established, the first missing visit due to withdrawal \(D_i=L_i+1\).
These checks prevent two common errors. First, equating one missing visit with dropout ignores the possibility of return. Second, equating a terminal sequence with withdrawal ignores other reasons follow-up may stop. Response indicators describe which outcomes were observed; documentation and the study schedule establish how the observation period ended.
Comparison, Limitations, and Common Pitfall
Temporary missed visits and permanent withdrawal are different observed-history patterns, but neither label explains the underlying cause. A participant can miss visits because of travel, illness, scheduling difficulty, or other circumstances and later return. A terminal pattern may reflect withdrawal, but it can also reflect administrative end or an unrelated event. The distinction is descriptive and operational; it is not, by itself, a claim about the missingness mechanism or about how outcomes should be analyzed.
| Feature | Temporary missed visit | Permanent withdrawal |
|---|---|---|
| Later scheduled outcome observed? | Yes, after at least one missed visit | No outcomes are collected after follow-up ends |
| Response pattern | Includes a zero followed by a one | Terminal zeros after the last observed outcome |
| Dropout interpretation | The missed visit is not terminal dropout | First missing visit due to withdrawal can be identified, if status is confirmed |
| What the pattern alone establishes | An outcome was missed and follow-up later resumed | Later outcomes are unobserved; the reason for the boundary still needs checking |
Common pitfall: “The participant has a missing outcome at visit 3, so dropout occurred at visit 3.” This reasoning mistakes a single zero for a terminal event. If the history is \((1,1,0,1,1)\), the visit-4 one proves that outcome collection resumed, so visit 3 is an intermittent missing visit, not dropout. Conversely, a history such as \((1,1,1,0,0)\) is consistent with dropout only when the study records establish withdrawal rather than another reason for the terminal boundary.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A participant’s response history is \((1,0,1,0,0)\). Is the visit-2 absence terminal dropout? Explain.
- In a seven-visit study, a participant is observed through visit 5 and formally withdraws before visit 6. Give \(L_i\) and \(D_i\).
- Why is \((1,1,1,0,0)\) not enough by itself to prove permanent withdrawal?
- A participant misses visits 2 and 3, then has observed outcomes at visits 4 and 5. How should the absences be described?
- State the distinction between the last observed visit and the first missing visit due to withdrawal.
Solutions. (1) No. \(R_{i2}=0\) is followed by \(R_{i3}=1\), so visit 2 is an intermittent missed visit. The later terminal zeros may be consistent with an eventual follow-up boundary, whose reason must be checked. (2) \(L_i=5\), and \(D_i=L_i+1=6\). (3) The terminal zeros describe unobserved outcomes but do not identify why observation ended; withdrawal must be distinguished from administrative end or another boundary. (4) They are intermittent missing visits because later outcomes were observed. (5) The last observed visit \(L_i\) is the final scheduled visit with an observed outcome; if withdrawal occurs before the next scheduled outcome, the first missing visit due to withdrawal is \(D_i=L_i+1\).